Iterative properties of birational rowmotion I: Generalities and skeletal posets

Darij Grinberg, Tom Roby

Research output: Contribution to journalArticlepeer-review

17 Scopus citations


We study a birational map associated to any finite poset P. This map is a far-reaching generalization (found by Einstein and Propp) of classical rowmotion, which is a certain permutation of the set of order ideals of P. Classical rowmotion has been studied by various authors (Fon-der-Flaass, Cameron, Brouwer, Schrijver, Striker, Williams and many more) under different guises (Striker-Williams promotion and Panyushev complementation are two examples of maps equivalent to it). In contrast, birational rowmotion is new and has yet to reveal several of its mysteries. In this paper, we set up the tools for analyzing the properties of iterates of this map, and prove that it has finite order for a certain class of posets which we call\skeletal”. Roughly speaking, these are graded posets constructed from one-element posets by repeated disjoint union and \grafting onto an antichain”; in particular, any forest having its leaves all on the same rank is such a poset. We also make a parallel analysis of classical rowmotion on this kind of posets, and prove that the order in this case equals the order of birational rowmotion.

Original languageEnglish (US)
JournalElectronic Journal of Combinatorics
Issue number1
StatePublished - Feb 19 2016

Bibliographical note

Publisher Copyright:
© 2016, Australian National University. All rights reserved.


  • Graded posets
  • Order ideals
  • Posets
  • Rowmotion
  • Trees
  • Tropicalization


Dive into the research topics of 'Iterative properties of birational rowmotion I: Generalities and skeletal posets'. Together they form a unique fingerprint.

Cite this